Q51. A and B can complete a work in 15 days, B and C in 20 days, and A and C in 30 days. In how many days can A, B, and C complete the work together?
Formula: (A+B) + (B+C) + (A+C) = 2(A+B+C)
Substitution: 1/15 + 1/20 + 1/30 = 2(A+B+C)
LCM = 60 → (4 + 3 + 2)/60 = 9/60 = 3/20
So, 2(A+B+C) = 3/20 → (A+B+C) = 3/40
Final Step: Time = 40/3 = 13⅓ days
Final Answer: 13⅓ days
Q52. A can do a work in 15 days and B in 25 days. They start together, but A leaves after 5 days. Find the total time to complete the work.
Step 1: Combined rate: 1/15 + 1/25 = 8/75
Step 2: Work in 5 days: 5×8/75 = 40/75 = 8/15
Step 3: Remaining work: 1 − 8/15 = 7/15
Step 4: B’s rate = 1/25: Time = (7/15) ÷ (1/25) = 175/15 = 11⅔ days
Total time = 5 + 11⅔ = 16⅔ days
Q53. 12 men can finish a work in 9 days. How many men are required to finish it in 6 days?
Formula: M₁D₁ = M₂D₂
12×9 = M₂×6 → M₂ = 18 men
Final Answer: 18 men
Q54. A can complete a job in 10 days, B is 25% less efficient than A. Find B’s time to complete the job.
Efficiency ratio: A : B = 100 : 75 = 4 : 3
Time ratio (inverse): 3 : 4
A’s time = 10 days → B’s time = (4/3)×10 = 13⅓ days
Final Answer: 13⅓ days
Q55. A and B can do a piece of work in 12 days. A alone can do it in 18 days. Find how long B alone will take.
Formula: 1/B = 1/Together − 1/A
= 1/12 − 1/18 = (3−2)/36 = 1/36
Final Answer: 36 days
Q56. A can complete a work in 8 days, B in 10 days, and C in 20 days. They all start together, but C leaves after 4 days. Find total time to complete the work.
Step 1: Combined rate (A+B+C) = 1/8 + 1/10 + 1/20 = (5+4+2)/40 = 11/40
Step 2: Work in 4 days: 4×11/40 = 11/10
Remaining work = 1 − 11/10 = −1/10 (means completed early!)
Conclusion: Work completed in less than 4 days since all together are fast.
If we calculate precisely: Time = (40/11) ≈ 3.64 days
Final Answer: ≈ 3 days 15 hours
Q57. A can do a work in 15 days, B can do it in 25 days. Together they work for 5 days and leave the rest to C, who finishes in 5 more days. How long will C take alone?
Step 1: A+B’s rate = 1/15 + 1/25 = 8/75
Work in 5 days = 5×8/75 = 40/75 = 8/15
Remaining = 1 − 8/15 = 7/15
C completes 7/15 in 5 days → rate = (7/15)/5 = 7/75
Time = 1 ÷ (7/75) = 75/7 = 10⅚ days
Final Answer: ≈ 10 days 20 hours
Q58. A can do a work in 12 days, B is 60% efficient as A. How long will B alone take?
Efficiency ratio = 100 : 60 = 5 : 3
Time ratio (inverse) = 3 : 5
If A = 12 → B = (5/3)×12 = 20 days
Final Answer: 20 days
Q59. A can do 3/4 of a work in 9 days. How long will A take to do the full work?
Total time = (9 × 1)/(3/4) = 9×4/3 = 12 days
Final Answer: 12 days
Q60. A and B can complete a work in 10 days and 15 days respectively. Find the time if both work together.
1/T = 1/10 + 1/15 = (3+2)/30 = 1/6
Final Answer: 6 days
Q61. A and B can do a piece of work in 12 days. A alone can do it in 18 days. They work together for 4 days; find the fraction of work left.
1/B = 1/12 − 1/18 = 1/36
Combined rate = 1/12
Work in 4 days = 4×1/12 = 1/3
Remaining = 1 − 1/3 = 2/3 of work
Final Answer: 2/3 of work remains
Q62. A can do a work in 10 days, B in 12 days, and C in 15 days. If they work together, how long will they take?
1/T = 1/10 + 1/12 + 1/15 = (6+5+4)/60 = 15/60 = 1/4
Final Answer: 4 days
Q63. A can finish a work in 10 days, B in 20 days. They work together for 4 days, after which A leaves. How long will B take to complete the rest?
Combined rate = 1/10 + 1/20 = 3/20
Work done in 4 days = 4×3/20 = 3/5
Remaining = 2/5; B’s rate = 1/20
Time = (2/5) ÷ (1/20) = 8 days
Final Answer: 8 days
Q64. A is twice as efficient as B. If A and B together can finish a work in 12 days, find A’s time alone.
Let B’s efficiency = 1 unit, A = 2 units → total = 3 units
Work = 3×12 = 36 units
A’s rate = 2 → Time = 36/2 = 18 days
Final Answer: 18 days
Q65. A can do a work in 24 days, B can do it 50% faster. Find B’s time.
B’s efficiency = 150% of A
Time ratio (inverse) = 100:150 → 3:2
A = 24 → B = (2/3)×24 = 16 days
Final Answer: 16 days
Q66. A can do a work in 12 days, B in 18 days. If they work together for 3 days and A leaves, how long will B take to finish?
Combined rate = 1/12 + 1/18 = 5/36
Work in 3 days = 15/36 = 5/12
Remaining = 7/12; B’s rate = 1/18
Time = (7/12) ÷ (1/18) = 10½ days
Final Answer: 10½ days
Q67. A can complete a job in 9 days, B in 6 days. They start together but A leaves after 3 days. Find time for B to finish the rest.
Combined rate = 1/9 + 1/6 = 5/18
Work in 3 days = 15/18 = 5/6
Remaining = 1/6; B’s rate = 1/6
Final Answer: 1 more day
Q68. 5 men can do a job in 8 days. How long will 10 men take?
M₁D₁ = M₂D₂ → 5×8 = 10×D₂
D₂ = 4 days
Final Answer: 4 days
Q69. A can do a work in 20 days, B in 30 days, and C in 60 days. Find total time if all work together.
1/T = 1/20 + 1/30 + 1/60 = (3+2+1)/60 = 6/60 = 1/10
Final Answer: 10 days
Q70. A does half of a work in 8 days. B completes the remaining half in 6 days. How long would both take together?
A’s full time = 8×2 = 16; B’s full time = 6×2 = 12
1/T = 1/16 + 1/12 = (3+4)/48 = 7/48
T = 48/7 = 6.85 days
Final Answer: ≈ 6 days 20 hours
Q71. A can do a work in 10 days, B in 15 days. Find B’s efficiency compared to A.
Efficiency ∝ 1/time → A:B = 15:10 = 3:2
Final Answer: B is 66.67% as efficient as A
Q72. A can do a job in 12 days, B can do it in 18 days. How long will both take if they work together for 3 days, rest 1 day, and repeat?
Combined rate = 1/12 + 1/18 = 5/36
Work in 3 days = 15/36 = 5/12 per 4-day cycle
Remaining work = 1 − 5/12 = 7/12
Second cycle adds 5/12 → remaining = 2/12
3rd cycle → need only 3×(2/12)/(5/12)=1.2 days
Total Time: 8 + 1.2 = 9.2 days (≈ 9 days 5 hours)
Q73. A and B can do a work in 8 days, A alone can do it in 12 days. Find A’s and B’s efficiency ratio.
1/B = 1/8 − 1/12 = 1/24 → A:B = (1/12):(1/24) = 2:1
Final Answer: A : B = 2 : 1
Q74. A can do a work in 20 days, B can do it in 25 days. If they work together for 5 days, what part remains?
1/T = 1/20 + 1/25 = 9/100
Work in 5 days = 45/100 = 9/20
Remaining = 1 − 9/20 = 11/20
Final Answer: 11/20 of work remains
Q75. A and B can do a work in 9 days, B and C in 12 days, and A and C in 18 days. Find A’s time alone.
(A+B)+(B+C)+(A+C)=2(A+B+C)
1/9 + 1/12 + 1/18 = 2(A+B+C)
LCM=36 → (4+3+2)/36 = 9/36 = 1/4
So (A+B+C)=1/8
A = (A+B+C)−(B+C)=1/8−1/12=1/24
Final Answer: 24 days
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